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physica status solidi (b)

On the Definition of the Current Operator Application to the Hubbard Model

Dario Cabib, T. A. Kaplan

Published Jul 1, 1973• 15 citations
Publisher

Abstract

Abstract A disagreement existing in the literature between various calculations of the electrical conductivity in the single‐band Hubbard model is resolved. The error, which occurs in a paper by Monecke, is pointed out and explained to be due to the assumption that the current operator, defined as the commutator of the Hubbard Hamiltonian with the polarization operator, is invariant under the lattice translations. The existence of ambiguity in the definition of a “good” current operator (which is translationally invariant) is pointed out and resolved in a large class of situations.

DOI
10.1002/pssb.2220580108

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On the Definition of the Current Operator Application to the Hubbard Model

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Abstract A disagreement existing in the literature between various calculations of the electrical conductivity y in the single‐band Hubbard model is resolved. It is replied to an article by Cabib and Kaplan who tried to disprove that there are no absorption and emission lines near ω = U due to the translational invariance. The physical reason for this result and some of its consequences are discussed.

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Physical review. B, Condensed matter•Apr 15, 1996•1 citation

A note on ferromagnetism in the Hubbard model on the complete graph

Andreas Mielke, Hal Tasaki

Recently there have appeared some papers which discuss the existence of ferromagnetism in the Hubbard model defined on the complete graph. At least for the special electron number N_e=N-1, where N denotes the number of sites in the lattice, the existence of ferromagnetism in this model was established rigorously some time ago, as special (and indeed the simplest) cases of more general classes of models. Here we explain these implications to clarify the situation, although we believe the implications are straightforward. We are posting this note to the preprint archive to make it public, but we are not planning to publish it in other forms. This is because we do not think the problem warrants any extra publications, and we believe that the validity of our explanation is evident to the readers.

arXiv:cond-mat/9606115•Jun 17, 1996•0 citations
Abstract A disagreement existing in the literature between various calculations of the electrical conductivity in the single‐band Hubbard model is resolved. The error, which occurs in a paper by Monecke, is pointed out and explained to be due to the assumption that the current operator, defined as the commutator of the Hubbard Hamiltonian with the polarization operator, is invariant under the lattice translations. The existence of ambiguity in the definition of a “good” current operator (which is translationally invariant) is pointed out and resolved in a large class of situations.
Abstract An approximate expression for the electrical conductivity σ is derived by means of the method of spectral moments. Therefore, σ is written in terms of a spectral density, for which a suitable ansatz is chosen. Free parameters of this ansatz are fixed by equating moments. The so obtained approximate solution of the conductivity tensor fulfils the well‐known sum rules and contains nearly all exactly calculable limiting cases. Furthermore, Moneckes statement that the conductivity exhibits free acceleration behaviour even in the case of strong electronic correlations, is shown to be incorrect.
This chapter shows how, for noninteracting particles, the charge and energy current density operators are easily obtained from the continuity equation, with the time derivative of the corresponding density operators given by their commutator with the Hamiltonian in the presence of the external fields. This can be generalized to include spin-orbit scattering from heavy impurities, which gives rise to a spin current. For interacting particles, the commutator is very hard to calculate, and the uniform current operators are derived from the rate of change of the charge and energy polarization operators, which are defined as the first moments of the charge and energy densities, respectively. This is done for continuous models, as well as for the Hubbard, periodic Anderson, and Falicov-Kimball lattice models.
The optical conductivity of the U=\ensuremath{\infty} one-band Hubbard model is calculated on a square lattice in leading order of a 1/N expansion. To achieve this, the X-operator formulation and a Baym-Kadanoff expansion are used. For N\ensuremath{\rightarrow}\ensuremath{\infty} the optical conductivity is given by a Drude weighted \ensuremath{\delta} function. The calculated Drude weight as a function of the density is in excellent agreement with recent results of an exact diagonalization study. This supports the limit N\ensuremath{\rightarrow}\ensuremath{\infty} as being a good starting point for practical calculations. \textcopyright{} 1996 The American Physical Society.
Recently there have appeared some papers which discuss the existence of ferromagnetism in the Hubbard model defined on the complete graph. At least for the special electron number N_e=N-1, where N denotes the number of sites in the lattice, the existence of ferromagnetism in this model was established rigorously some time ago, as special (and indeed the simplest) cases of more general classes of models. Here we explain these implications to clarify the situation, although we believe the implications are straightforward. We are posting this note to the preprint archive to make it public, but we are not planning to publish it in other forms. This is because we do not think the problem warrants any extra publications, and we believe that the validity of our explanation is evident to the readers.